I got a cheap Bolens push mower from the big blue store a couple of years ago. It has a Briggs engine, but no bag, just a side chute. At the time it was well under $200, but the same mower is not far from $300 now. I use the push mower for areas too tight for my lawn tractor.
This cheapie has been one of the best push mowers I've ever owned, and it replaced a piece-of-garbage Craftsman I got from a local Sears that was going out of business. The pull start on the Bolens is the easiest I've ever used, and believe me, I've dealt with some real bears. This mower starts right up, no drama, no complaints.
You might be better off with something like this instead of trying to keep that Vietnamesium crap going. The problem with that stuff is the same as I've run into with Chinese stuff: essentially, pot metal used everywhere it should never be.